PolynomialNormalForm - Maple Help

DEtools

 PolynomialNormalForm
 construct the differential polynomial normal form of a rational function

 Calling Sequence PolynomialNormalForm(F, x)

Parameters

 F - rational function of x x - variable

Description

 • Let F be a rational function of x over a field K of characteristic 0. The PolynomialNormalForm(F,x) command constructs the differential polynomial normal form for F.
 • The output is a sequence of 3 elements $a,b,c$ where $a,b,c$ are polynomials over K such that:
 1 $F=\frac{a}{b}+\frac{\frac{ⅆ}{ⅆx}c}{c}.$
 2 $\mathrm{gcd}\left(b,a-i\frac{{\partial }}{{\partial }x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}b\right)=1$ for all non-negative integers $i$.
 3 $\mathrm{gcd}\left(b,c\right)=1$.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $F≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^{2}}-\frac{9}{{\left(x-1\right)}^{2}}-\frac{9{x}^{2}+12}{{x}^{3}+4x-2}+\frac{1}{{\left({x}^{3}+4x-2\right)}^{2}}$
 ${F}{≔}\frac{{4}}{{x}{-}{2}}{+}\frac{{4}}{{x}{+}{1}}{-}\frac{{3}}{{\left({x}{+}{1}\right)}^{{2}}}{-}\frac{{9}}{{\left({x}{-}{1}\right)}^{{2}}}{-}\frac{{9}{}{{x}}^{{2}}{+}{12}}{{{x}}^{{3}}{+}{4}{}{x}{-}{2}}{+}\frac{{1}}{{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}}$ (1)
 > $a,b,c≔\mathrm{PolynomialNormalForm}\left(F,x\right)$
 ${a}{,}{b}{,}{c}{≔}{-}{5}{}{{x}}^{{9}}{-}{16}{}{{x}}^{{8}}{-}{14}{}{{x}}^{{7}}{-}{134}{}{{x}}^{{6}}{+}{39}{}{{x}}^{{5}}{-}{331}{}{{x}}^{{4}}{+}{96}{}{{x}}^{{3}}{+}{32}{}{{x}}^{{2}}{+}{16}{}{x}{-}{7}{,}{\left({x}{+}{1}\right)}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({{x}}^{{3}}{+}{4}{}{x}{-}{2}\right)}^{{2}}{,}{\left({x}{-}{2}\right)}^{{4}}$ (2)

Check the result:

 > $\mathrm{nF}≔\frac{a}{b}+\frac{\mathrm{diff}\left(c,x\right)}{c}:$
 > $\mathrm{Testzero}\left(\mathrm{normal}\left(F-\mathrm{nF}\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{res}≔\mathrm{resultant}\left(b,a-j\mathrm{diff}\left(b,x\right),x\right):$
 > $H≔\mathrm{select}\left(\mathrm{type},\left\{\mathrm{solve}\left(\mathrm{res},j\right)\right\},'\mathrm{nonnegint}'\right):$
 > $\mathrm{evalb}\left(H=\varnothing \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{gcd}\left(b,c\right)=1\right)$
 ${\mathrm{true}}$ (4)

References

 Almkvist, G, and Zeilberger, D. "The method of differentiating under the integral sign." Journal of Symbolic Computation. Vol. 10. (1990): 571-591.